Theorem exmoeu2 1040

Description: Existence implies "at most one" is equivalent to uniqueness.

Assertion

Ref	Expression
exmoeu2	⊢ (∃xφ → (∃*xφ ↔ ∃!xφ))

Proof of Theorem exmoeu2

Step	Hyp	Ref	Expression
1		eu5 1035	. 2 ⊢ (∃!xφ ↔ (∃xφ ∧ ∃*xφ))
2	1	baibr 507	1 ⊢ (∃xφ → (∃*xφ ↔ ∃!xφ))

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678  ∃!weu 1007  ∃*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org